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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "paths/standard_precedence.ma".
17 (* STANDARD ORDER ************************************************************)
19 (* Note: this is p ≤ q *)
20 definition sle: relation path ≝ star … sprec.
22 interpretation "standard 'less or equal to' on paths"
25 lemma sle_step_rc: ∀p,q. p ≺ q → p ≤ q.
29 lemma sle_step_sn: ∀p1,p,p2. p1 ≺ p → p ≤ p2 → p1 ≤ p2.
33 lemma sle_rc: ∀p,q. dx::p ≤ rc::q.
37 lemma sle_sn: ∀p,q. rc::p ≤ sn::q.
41 lemma sle_skip: dx::◊ ≤ ◊.
45 lemma sle_nil: ∀p. ◊ ≤ p.
49 lemma sle_comp: ∀p1,p2. p1 ≤ p2 → ∀o. (o::p1) ≤ (o::p2).
50 #p1 #p2 #H elim H -p2 // /3 width=3/
53 lemma sle_skip_sle: ∀p. p ≤ ◊ → dx::p ≤ ◊.
54 #p #H @(star_ind_l … p H) -p //
55 #p #q #Hpq #_ #H @(sle_step_sn … H) -H /2 width=1/
58 theorem sle_trans: transitive … sle.
62 lemma sle_cons: ∀p,q. dx::p ≤ sn::q.
64 @(sle_trans … (rc::q)) /2 width=1/
67 lemma sle_dichotomy: ∀p1,p2:path. p1 ≤ p2 ∨ p2 ≤ p1.
70 | #o1 #p1 #IHp1 * /2 width=1/
71 * #p2 cases o1 -o1 /2 width=1/
72 elim (IHp1 p2) -IHp1 /3 width=1/
76 lemma sle_inv_dx: ∀p,q. p ≤ q → ∀q0. dx::q0 = q →
77 in_whd p ∨ ∃∃p0. p0 ≤ q0 & dx::p0 = p.
78 #p #q #H @(star_ind_l … p H) -p [ /3 width=3/ ]
79 #p0 #p #Hp0 #_ #IHpq #q1 #H destruct
80 elim (IHpq ??) -IHpq [4: // |3: skip ] (**) (* simplify line *)
81 [ lapply (sprec_fwd_in_whd … Hp0) -Hp0 /3 width=1/
82 | * #p1 #Hpq1 #H elim (sprec_inv_dx … Hp0 … H) -p
83 [ #H destruct /2 width=1/
89 lemma sle_inv_rc: ∀p,q. p ≤ q → ∀p0. rc::p0 = p →
90 (∃∃q0. p0 ≤ q0 & rc::q0 = q) ∨
92 #p #q #H elim H -q /3 width=3/
93 #q #q0 #_ #Hq0 #IHpq #p0 #H destruct
94 elim (IHpq p0 ?) -IHpq // *
96 elim (sprec_inv_rc … Hq0 … H) -q * /3 width=3/ /4 width=3/
97 | #p1 #H elim (sprec_inv_sn … Hq0 … H) -q /3 width=2/
101 lemma sle_inv_sn: ∀p,q. p ≤ q → ∀p0. sn::p0 = p →
102 ∃∃q0. p0 ≤ q0 & sn::q0 = q.
103 #p #q #H elim H -q /2 width=3/
104 #q #q0 #_ #Hq0 #IHpq #p0 #H destruct
105 elim (IHpq p0 ?) -IHpq // #p1 #Hp01 #H
106 elim (sprec_inv_sn … Hq0 … H) -q /3 width=3/
109 lemma in_whd_sle_nil: ∀p. in_whd p → p ≤ ◊.
110 #p #H @(in_whd_ind … H) -p // /2 width=1/
113 theorem in_whd_sle: ∀p. in_whd p → ∀q. p ≤ q.
114 #p #H @(in_whd_ind … H) -p //
115 #p #_ #IHp * /3 width=1/ * #q /2 width=1/
118 lemma sle_nil_inv_in_whd: ∀p. p ≤ ◊ → in_whd p.
119 #p #H @(star_ind_l … p H) -p // /2 width=3 by sprec_fwd_in_whd/
122 lemma sle_inv_in_whd: ∀p. (∀q. p ≤ q) → in_whd p.
123 /2 width=1 by sle_nil_inv_in_whd/
126 lemma sle_fwd_in_whd: ∀p,q. p ≤ q → in_whd q → in_whd p.
127 #p #q #H @(star_ind_l … p H) -p // /3 width=3 by sprec_fwd_in_whd/
130 lemma sle_fwd_in_inner: ∀p,q. p ≤ q → in_inner p → in_inner q.
131 /3 width=3 by sle_fwd_in_whd/